Analytical comparison between the surface current integral equation and the second-order small-slope approximation

Abstract

This paper is the third in a series discussing a new approximate bistatic model for electromagnetic scattering from perfectly conducting rough surfaces. Our previous approach supplemented the Kirchhoff model through the addition of new terms involving linear orders in slope and surface elevation differences that arise naturally from a second iteration of the surface current integral equation. This completion of the Kirchhoff was shown to provide the correct first-order small perturbation method (SPM-1) in the general bistatic context. The agreement with SPM-1 was achieved because differences of surface heights are no longer expanded in powers of surface slope. While consistent with SPM, our previous formulation fails to reconverge toward the Kirchhoff model, at some incidence and scattered angles, when the illuminated surface satisfies the high frequency roughness condition. This weakness is also shared with the first-order small slope approximation (SSA-1) which is structurally equivalent to our previous formulation where the polarization is independent of surface roughness. The second-order small slope approximation (SSA-2), which satisfies the SPM-1 and second-order small perturbation method (SPM-2) limits by construction, was shown by Voronovich to converge toward the tangent plane approximation of the Kirchhoff model under high frequency conditions. In the present paper, we show that, in addition to the linear orders in our previous model, one must now include cross-terms between slope and surface elevation to ensure convergence toward both high frequency and small perturbation limits. With the inclusion of these terms, our new formulation becomes comparable to the SSA-2 (second-order kernel) without the need to evaluate all the quadratic order slope and elevations terms. SSA-2 is more complete, however, in the sense that it guarantees convergence toward the second-order Bragg limit (SPM-2) in the fully dielectric case in addition to both SPM-1 and Kirchhoff. Our new generalization is shown to explain correctly extra depolarization in specular conditions to be caused by surface curvature and surface autocorrelation for incoherent and coherent scattering, respectively. This result will have large repercussions on the interpretation of bistatically reflected signals such as those from GPS.     

On the high-frequency limit of the second-order small-slope approximation

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

On the high-frequency limit of the second-order small-slope approximation

Abstract

Small-slope approximation (SSA) is a scattering theory that is supposed to unify both the small-perturbation model and the Kirchhoff approximation (KA). We study and compute the second-order small-slope approximation (SSA2) in a high-frequency approximation (SSA2-hf) that makes it proportional to the first-order term, with a roughness-independent factor. For the 3D electromagnetic problem we show analytically that SSA2-hf actually meets KA in the case of perfectly conducting surfaces. This no longer holds in the dielectric case but we give numerical evidence that the two methods remain extremely close to each other for moderate scattering angles. We discuss the potential applications of SSA2-hf and give some 2D numerical comparison with rigorous computations.     

An extension of the small-slope approximation for rough surface scattering

Solutions are derived for scattering from a rough one-dimensional pressure-release surface in the form of a functional series in the surface slope. These solutions are obtained by solving an integral equation of the first kind for the surface potential to obtain a representation for the scattering amplitude. It is shown that the subsequent expansion of terms occurring in the scattering amplitude to obtain a functional series in the slope does not yield a unique result. The result obtained contains a free parameter that may be arbitrarily selected. Thus, this result is an extension or generalization of the small-slope approximation of Voronovich (1985 Sov. Phys.-JETP 62 65-70) that differs at second order in the slope from his result. It is also shown that the free parameter can be selected such that each term of the functional series is reciprocal and exhibits a limiting grazing angle dependence consistent with the requirements of flux conservation and the absence of boundary waves. A new formulation of the leading terms of the small-slope expansions is derived and is used to explore the conditions under which the two expansions reduce to the Kirchhoff approximation. Finally, a numerical example is presented to demonstrate that the extended approximation provides corrections that are important for near grazing scatter.     

A study of the higher-order small-slope approximation for scattering from a Gaussian rough surface

Results from the first three terms of the small-slope approximation (SSA) for incoherent electromagnetic scattering from a penetrable randomly rough interface are discussed. Surface roughness is characterized as a Gaussian random process with an isotropic Gaussian correlation function. Sample results illustrate parameter spaces for which each correction term is appreciable. Reduction of the SSA to the physical optics theory is also discussed for both perfectly conducting and dielectric surfaces.     

Molten salts near a charged surface: integral equation approximation for a model of KCl

A study has been made of the electrical ‘double layer’ structure of molten salts, in particular a model of molten potassium chloride, using an integral equation approximation. This is in contrast to most statistical mechanical treatments of the double layer, which have concentrated on aqueous elecrolyte solutions. The results are compared with the output of computer simulations. In addition to the structural information contained in the density profiles, the calculations yielded charge profiles, the mean electrostatic potential and the double layer capacitance.     

The Fokker-Planck equation in the second-order pitch angle approximation and its exact solution

The diffusive particle propagation and its pitch angle scattering is studied using kinetic equation of the Fokker-Planck form. The case is considered when charged particles preferable propagate along the strong mean magnetic field direction and undergo the pitch angle scattering with respect to it. The paper deals with solution of the equation for particle distribution function in the second-order approximation in the pitch angle. The exact analytical solution is obtained in an integral form. The well-known solution in the first-order pitch angle approximation can be restored performing the small time limit in the result. Unlike the first-order solution the obtained solution in the second approximation rightly shows that the pitch angle diffusion is closely connected with the particle transport along the mean magnetic field. The expression for particle density for the point instantaneous unidirectional source also has been obtained.     

The effect of the modulation of Bragg scattering in small-slope approximation

Small-slope approximation (SSA) belongs to a class of 'unifying' scattering theories which reproduce small perturbations and semiclassic (Kirchhoff) results within appropriate limits. However, the most stringent test for such theories involves a two-scale situation when a small-scale roughness is located on a tilted plane. A 'unifying' theory should properly account for the effects of modulation of the scattering cross section associated with a large-scale tilt. This paper shows that SSA does properly take into account these modulation effects.     

Analytical solution of Everett integral using Lorentzian Preisach function approximation

In this paper, we take in to account the Lorentzian function, already proposed as analytical function approximation in scalar Preisach-type modeling of hysteresis of soft magnetic materials. Preliminarily, we point out the properties of the Lorentzian function and the physical and mathematical meaning of its parameters. Successively, we show how the use of the Lorentzian function approximation allows to solve in complete analytical way the Everett's integral. In particular, we present in the paper the analytical expression in closed form of the first magnetization curve, of the symmetric and non-symmetric minor loops, and of the first- and second-order reversal curves. In addition, we show the use of the complete analytical formulas of the symmetric magnetic loops above-mentioned, applied to a simple identification procedure of the Lorentzian function parameters, by the knowledge of the measured major loop. Finally, in order to show the practical use of the analytical expression found, some computation examples and comparisons with experimental data are shown.     

The effect of the modulation of Bragg scattering in small-slope approximation

Abstract

Small-slope approximation (SSA) belongs to a class of ‘unifying’ scattering theories which reproduce small perturbations and semiclassic (Kirchhoff) results within appropriate limits. However, the most stringent test for such theories involves a two-scale situation when a small-scale roughness is located on a tilted plane. A ‘unifying’ theory should properly account for the effects of modulation of the scattering cross section associated with a large-scale tilt. This paper shows that SSA does properly take into account these modulation effects.     

Fluid in contact with semipermeable vesicle: second-order integral equation approach

Secondorder Ornstein–Zernike integral equations in conjunction with the Lovett–Mou–Buff–Wertheim equation for the density profile are used to investigate a mixture of hard spheres in contact with a semipermeable membrane of spherical symmetry. Theoretical predictions are compared with grand canonical Monte Carlo simulations for several parameters, and reasonable agreement has been found. The pair functions for the systems considered are also determined and discussed.     

Analytical solution of the double-logarithmic integral equation with QCD running coupling

The analytical solution of the double-logarithmic integral equation with QCD running coupling describing small-x behaviour of the non-singlet structure function ? _NS(x,Q ²) has been found for any cut-off parameter μ. Analytical properties of the solution and a position of the right-most singularity in the complex ρ-plane which determines the asymptotics of ? _NS(x,Q ²) at small x have been studied. The asymptotical formula ? _NS(x,Q ²) = C ₁ x ^-λ1{ln^κ1(Q ²/Λ²) —ln^κ1 (μ ²/Λ²) + κ ₁ ln^κ1-1(Q ²/Λ²)[ψ(1) - ψ(λ₁)]} valid if x ? 1 and ln(Q ²/Λ²) ? 1 has been obtained where C ₁, λ₁ are constants, κ ₁ = g/λ₁, λ₁ < g = 8/(33 - 2gh _f), gh _f is a number of active flavours and ψ(ξ) denotes the digamma function.     

A two-scale model from the point of view of the small-slope approximation

Abstract

General results for the scattering cross section following from the small-slope approximation (SSA) are applied to the case of two-scale surface roughness which can be represented as a superposition of small-scale and large-scale components. The purpose of the paper is to obtain analytically tractable results with obvious physical meaning which can be used for estimates prior to undertaking extensive numerical calculations according to exact unambiguous expressions of the SSA. The general case of vector (electromagnetic) or scalar (sound) waves is considered. The statistics of small-scale roughness is not assumed to be Gaussian (in any sense) or space-homogeneous, and the possible dependence of the statistics of small-scale roughness on a large-scale undulating surface is taken into account. As a result, a modified local spectrum of small-scale components of roughness enters into corresponding expressions for the scattering cross section. It is demonstrated that under appropriate conditions, the resulting formulae for the scattering cross section reduce to the conventional two-scale model.     

Scattering by two-dimensional rough surfaces: comparison between the method of moments,Kirchhoff and small-slope approximations

Abstract

We use a rigorous numerical code based on the method of moments to test the accuracy and validity domains of two popular first-order approximations, namely the Kirchhoff and the small-slope approximation(SSA), in the case of two-dimensional rough surfaces. The experiment is performed on two representative types of surfaces: surfaces with Gaussian spectrum, which are the paradigm of single-scale surfaces, and ocean-like surfaces, which belong to the family of multi-scale surfaces. The main outcome of these computations in the former case is that the SSA is outperformed by the Kirchhoff approximation(KA) outside the near-perturbative domain and in fact is quite unpredictable in that its accuracy does not depend only on the slope. For ocean-like surfaces, however, SSA behaves surprisingly well and is more accurate than the KA.     

Scattering by two-dimensional rough surfaces: comparison between the method of moments, Kirchhoff and small-slope approximations

We use a rigorous numerical code based on the method of moments to test the accuracy and validity domains of two popular first-order approximations, namely the Kirchhoff and the small-slope approximation(SSA), in the case of two-dimensional rough surfaces. The experiment is performed on two representative types of surfaces: surfaces with Gaussian spectrum, which are the paradigm of single-scale surfaces, and ocean-like surfaces, which belong to the family of multi-scale surfaces. The main outcome of these computations in the former case is that the SSA is outperformed by the Kirchhoff approximation(KA) outside the near-perturbative domain and in fact is quite unpredictable in that its accuracy does not depend only on the slope. For ocean-like surfaces, however, SSA behaves surprisingly well and is more accurate than the KA.     

Analytical solution to the material balance equation by integral transform for different cathode geometries

The 1-dimensional non-homogeneous material balance equation has been examined in the rectangular, spherical and cylindrical coordinate system. The solutions to this equation in the respective coordinate system has been determined analytically using the method of integral transform, together with the norms and eigen values suitable for galvanostatic discharge boundary conditions.     

Array current reconstruction by surface wave measurements. Two-dimensional approximation

The dependencies of the array current on the underlying surface properties were studied in the two-dimensional approximation. The results of the array current reconstruction were compared with experimental data of georadar sounding, obtained on the water basin surface. An array current reconstruction algorithm most adequately consistent with experimental data was obtained.